Internal Systemic Risk

Internal systemic risk is the uncertainty that arises because the actuarial valuation models are an imperfect representation of the insurance process: risk internal to the valuation process itself, also called model specification risk. In Marshall et al. (2008) it is one of the three sources of uncertainty behind a risk margin. Stochastic models fitted to the data cannot see it, so it is assessed by scoring the modelling infrastructure on a balanced scorecard and mapping the score to a CoV.

$\bar s = \frac{\sum_r v_r\,s_r}{\sum_r v_r}$

$\mathrm{CoV}_{\text{int}} = \frac{\sqrt{\sum_i\sum_j \rho_{ij}\,(w_i c_i)(w_j c_j)}}{\sum_i w_i}$

$s_r$ is risk indicator $r$ 's score from $1$ to $5$ ( $5$ is best practice), $v_r$ its weight and $\bar s$ the valuation class's weighted average score. A CoV scale maps $\bar s$ to the class's CoV $c_i$ . Classes are then combined using their shares of the central estimate $w_i$ and correlations $\rho_{ij}$ ( $\rho_{ii} = 1$ ).
The three sources, each with its own risk indicators:
- Specification error is the inability to build a model fully representative of the insurance process. Indicators include the number of independent models, whether claim or payment types are modelled separately, the range of results, reasonableness checks, goodness of fit, subjective adjustments, monitoring, the ability to detect trends, and the sophistication of the superimposed inflation analysis.
- Parameter selection error arises because the model cannot adequately measure all the predictors of claim cost outcomes, or trends in them. Indicators include whether the best predictors have been identified, whether they are stable over time, and whether those used lead rather than lag claim costs.
- Data error comes from poor or unavailable data, or inadequate knowledge of the portfolio, including its pricing, underwriting and claims processes. Indicators include knowledge of past processes, information from the business, reconciliations, robust data processes, and past revisions of data.
Scoring honestly. Each indicator is scored against best practice even where best practice is not possible for the portfolio, since it is easy to be defensive about one's own models. Weights reflect how much each indicator matters for that class. The scoring is of the approach actually used for the central estimate.
The CoV scale comes from judgement, supported by hindsight analysis of past model performance. Marshall et al. suggest the minimum CoV for a "perfect" model is unlikely to be much less than $5\%$ . For a single aggregated model with limited data, few predictors and heavy subjective adjustment, $20\%$ or more is readily justifiable and may be the largest source of uncertainty. Their example scale is not linear, and is higher for long-tail classes.
Correlation. It is uncorrelated with independent and external systemic risk. It is correlated between valuation classes, through the "same actuary" effect and template models, and between outstanding claim and premium liabilities where the premium liability method uses outstanding claim results.

From a Balanced Scorecard to a CoV A long-tail liability class is scored on six indicators:

Component Indicator Score Weight

Specification Number of independent models used $3$ $5$

Specification Checks made on reasonableness of results $4$ $4$

Specification Superimposed inflation analysis $2$ $6$

Parameter selection Value of predictors used $3$ $5$

Data Reconciliations and quality control $5$ $3$

Data Past mis-estimation from data revision $4$ $2$

The CoV scale for this class (the paper's example scale for its long-tail CTP class) is $11.5\%$ for scores of $3.0$ to $3.5$ and $9.5\%$ for $3.5$ to $4.0$ . Find the internal systemic risk CoV, and the effect of improving the superimposed inflation analysis to a score of $4$ .
Answer
 $\begin{align*} \bar s &= \frac{3(5) + 4(4) + 2(6) + 3(5) + 5(3) + 4(2)}{5 + 4 + 6 + 5 + 3 + 2} \\ &= \frac{81}{25} \\ &= 3.24 \end{align*}$ 
The score falls in the $3.0$ to $3.5$ band, so the CoV is $11.5\%$ . With the superimposed inflation indicator at $4$ , the numerator rises by $2 \times 6 = 12$ :
 $\begin{align*} \bar s &= \frac{93}{25} \\ &= 3.72 \end{align*}$ 
This moves the class into the next band and cuts the CoV to $9.5\%$ . A heavily weighted indicator scored poorly is where better analysis pays off most.

Component	Indicator	Score	Weight
Specification	Number of independent models used	$3$	$5$
Specification	Checks made on reasonableness of results	$4$	$4$
Specification	Superimposed inflation analysis	$2$	$6$
Parameter selection	Value of predictors used	$3$	$5$
Data	Reconciliations and quality control	$5$	$3$
Data	Past mis-estimation from data revision	$4$	$2$

Correlating Internal Systemic Risk Across Classes Liability makes up $60\%$ of outstanding claims, with an internal systemic risk CoV of $9\%$ , and property $40\%$ , with $6\%$ . The same actuary values both with similar methods, and the correlation is set in the medium band at $50\%$ .

Compute the combined CoV, and compare it with nil and full correlation.
Answer The weighted CoVs are $0.6 \times 9\% = 5.4\%$ and $0.4 \times 6\% = 2.4\%$ .
 $\begin{align*} \mathrm{CoV}^2 &= 0.054^2 + 0.024^2 + 2(0.5)(0.054)(0.024) \\ &= 0.002916 + 0.000576 + 0.001296 \\ &= 0.004788 \\ \mathrm{CoV} &= 6.92\% \end{align*}$ 
With nil correlation the CoV would be $\sqrt{0.003492} = 5.91\%$ ; with full correlation, $5.4\% + 2.4\% = 7.8\%$ . The shared actuary and methods keep most of the diversification from appearing, since a flaw in the approach would hit both classes at once.